Deep Symbolic Learning: Discovering Symbols and Rules from Perceptions⋆

Abstract

Neuro-Symbolic (NeSy) integration combines symbolic reasoning with Neural Networks (NNs) for tasks requiring both perception and reasoning. Most NeSy systems rely on continuous relaxation of logical knowledge, and they assume the symbolic rules to be given. We propose Deep Symbolic Learning (DSL) [1], a NeSy system that simultaneously learns the perception and symbolic functions while being trained only on their composition. The key idea is to adapt reinforcement learning policies to the NeSy context: given the NN predictions t with ∑︀ = 1, we use the greedy policy (t) = to select a single discrete symbol, and the function (t) = to select the corresponding value in t, interpreted as a truth value under a fuzzy logic semantics. When performing multiple discrete choices within the model, each truth value is sent to an aggregation operator, which returns the truth value of the final output.

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In Fig. 1, DSL architecture for the MNIST-Addition task [ 2 ] is provided. Two NNs, 1 and 2, are used to classify the images, and the softmax function converts their predictions into fuzzy truth values t1 and t2. The greedy policy function , takes in t1 and t2, and returns discrete symbolic predictions (8 and 0, resp.). Their corresponding fuzzy truth values (*1 = 0.5 and *2 = 0.6 resp.) are given by the operator . These symbols (8 and 0) are then passed to the symbolic function, which is represented as a lookup table G, returning the final output. The confidence in the final output is given by the confidence in the predicted symbols being simultaneously correct, using the Gödel semantics of conjunction, i.e., t* = (t*1, t*2). In Fig. 1, the NNs predict symbols 8 and 0, while the function defined by G corresponds to the sum. As a consequence, the final output of DSL is 8 (8 + 0 = 8). The framework considers the correctness of the final output, producing a label = 1 if the prediction is correct and = 0 otherwise. Such a label is given as supervision for t* . In the example, the prediction is wrong since the first digit has been classified as 8 instead of 3. Since the function admits only one non-zero partial derivative (corresponding to the minimum value), the back-propagation changes the weights of a single network (1 in Fig. 1). If the prediction is wrong, the efect of this change is to reduce the truth value of the currently selected symbol (8), increasing the chances of choosing a diferent symbol in the next iteration. Finally, DSL can also learn the table G from the data by applying and to a learnable tensor W.

In Tab. 1 we report the accuracy on the MNIST MultiDigitSum (MDS). For further experiments/analysis, see [ 1 ].

NAP [ 3 ] DPL [ 2 ] DStL[ 4 ] DSL

Acknowledgements TC and LS acknowledge the support of the PNRR project FAIR - Future AI Research (PE00000013), under the NRRP MUR program funded by the NextGenerationEU.

References

[1] A. Daniele , T. Campari , S. Malhotra , L. Serafini, Deep symbolic learning: Discovering symbols and rules from perceptions , arXiv preprint arXiv:2208.11561 (Accepted at IJCAI2023) ( 2023 ).

[2] R. Manhaeve , S. Dumancic , A. Kimmig , T. Demeester , L. De Raedt, Deepproblog: Neural probabilistic logic programming , NIPS ( 2018 ).

[3] Z. Yang , A. Ishay , J. Lee , Neurasp: Embracing neural networks into answer set programming , in: IJCAI , 2020 .

[4] T. Winters , G. Marra, R. Manhaeve , L. De Raedt, Deepstochlog: Neural stochastic logic programming , in: AAAI , 2022 .